Author: Brenda Gunderson, Ph.D., 2012
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Mind on Statistics
Utts/Heckard, 4th Edition, Cengage L, 2012
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Stat 250 Gunderson Lecture Notes
Learning about a Population Mean
Part 2: Confidence Interval for a Population Mean
Chapter 11: Sections 1 and 2, CI Module 3
Do not put faith in what statistics say until you have carefully considered
what they do not say. --William W. Watt
11.1 Introduction to Confidence Intervals for Means
Chapter 10 introduced us to confidence intervals for estimating a population
proportion and the difference between two population proportions. Recall it is
important to understand how to interpret an interval and how to interpret what
the confidence level really means.

The interval provides a range of reasonable values for the parameter with an
associated high level of confidence. For example we can say, “We are 95%
confident that the proportion of Americans who do not get enough sleep at
night is somewhere between 0.325 to 0.395, based on a random sample of n =
935 American adults.

The 95% confidence level describes our confidence in the procedure we used
to make the interval. If we repeated the procedure many times, we would
expect about 95% of the intervals to contain the population parameter.
11.2 CI Module 3: Confidence Interval for a Population
Mean 
Consider a study on the design of a highway sign. A question of interest is: What
is the mean maximum distance at which drivers are able to read the sign?
Data: A highway safety researcher will take a random sample of n = 16 drivers and
measure the maximum distances (in feet) at which each can read the sign.
Population parameter:
 = _________ mean maximum distance to read the sign for _____________
155
Sample estimate
x = _____ mean maximum distance to read the sign for _________________
But we know the sample estimate x may not equal  , in fact, the possible x
values vary from sample to sample. Because the sample mean is computed from
a random sample, then it is a random variable, with a probability distribution.
Sampling Distribution of the sample mean
If x is the sample mean for a random sample of size n from a population with a
normal model, then the distribution of the sample mean is:
Central Limit Theorem
If x is the sample mean for a random sample of size n from a population with any
model, with mean,  , and standard deviation  , then when n is large,
then the sampling distribution of the sample mean is approximately:
So the possible x values vary normally around  with a standard deviation of

n
. The standard deviation of the sample mean,  , is roughly the average
n
distance of the possible sample mean values from the population mean  . Since
we don’t know the population standard deviation  we will use the sample
standard deviation s, resulting in the standard error of the sample mean.
Standard Error of the Sample mean
s.e.( x ) =
where s = sample standard deviation
The standard error of x estimates, roughly, the average distance of the possible
x values from  The possible x values result from considering all possible
random samples of the same size n from the same population.
156
So we have our estimate of the population mean, the sample mean x , and we
have its standard error. To make our confidence interval, we need to know the
multiplier.
Sample Estimate  Multiplier x Standard error
The multiplier for a confidence interval for the population mean is denoted by
t*, which is the value in a Student’s t distribution with df = n – 1 such that the
area between –t and t equals the desired confidence level. The value of t* will be
found using Table A.2. First let’s give the formal result.
One-sample t Confidence Interval for 
x  t *s.e.( x )
where t * is an appropriate value for a t(n – 1) distribution.
This interval requires we have a random sample from a normal population. If
the sample size is large (n > 30), the assumption of normality is not so crucial and
the result is approximate.
Important items:
 be sure to check the conditions
 know how to interpret the confidence interval
 be able to explain what the confidence level of say 95% really means
Try It! Using Table A.2 to find t*
(a) Find t * for a 90% confidence interval based on n = 12 observations.
(b) Find t * for a 95% confidence interval based on n = 30 observations.
(c) Find t * for a 95% confidence interval based on n = 54 observations.
(d) What happens to the value of t * as the sample size (and thus the degrees of
freedom) gets larger?
157
From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition. 2012. Used with
permission.
158
Try It! Confidence Interval for the Mean Maximum Distance
Recall the study on the design of a highway sign. The researcher wanted to learn
about the mean maximum distance at which drivers are able to read the sign.
The researcher took a random sample of n = 16 drivers and measured the
maximum distances (in feet) at which each can read the sign. The data are
provided below.
440
360
490
600
600
490
540
400
540
490
600
540
240
440
440
490
a. Verify the necessary conditions for computing a confidence interval for the
population mean distance. We are told that the sample was a random sample
so we just need to check if a normal model for the response ‘max distance’ for
the population is reasonable.
Comments:
159
b. Compute the sample mean maximum distance and the standard error
(without the outlier).
c. Use a 95% confidence interval to estimate the population mean maximum
distance at which all drivers can read the sign. Write a paragraph that
interprets this interval and the confidence level.
One-Sample Statistics
N
DISTANCE
15
Mean
497.3333
Std. Deviation
73.43283
Std. Error
Mean
18.96028
One-Sample Test
Tes t Value = 0
DISTANCE
t
26.230
df
14
Sig. (2-tailed)
.000
160
Mean
Difference
497.3333
95% Confidence
Interval of the
Difference
Lower
Upper
456.6676 537.9991
Here is the summary of the Confidence intervals for the big 5 parameters covered
in Chapters 10 and 11. We have now covered the one and two population
proportion scenarios and the one population mean scenario
Population
Proportion
Parameter
Two Population Proportions
Parameter
p
p1  p 2
Parameter
pˆ 1  pˆ 2
Statistic
Standard Error
p̂
Statistic
Standard Error
s.e.( pˆ ) 
Population Mean
pˆ (1  pˆ )
n
s.e.( pˆ 1  pˆ 2 ) 
Confidence Interval
x
Statistic
Standard Error
pˆ 1 (1  pˆ 1 ) pˆ 2 (1  pˆ 2 )

n1
n2
Confidence Interval
pˆ  z s.e.( pˆ )
pˆ 
 pˆ 1  pˆ 2   z *s.e. pˆ 1  pˆ 2 
s
n
Paired Confidence
Interval
*
d  t s.e.(d ) df = n – 1
z*
2 n
 z* 

Sample Size n  

 2m 
s.e.( x ) 
Confidence Interval
x  t *s.e.( x ) df = n – 1
*
Conservative Conf.
Interval
2
Two Population Means
General
Parameter
Statistic
Standard Error
s.e.x1  x2  
s12 s22

n1 n2
1   2
x1  x 2
Pooled
x1  x2   t s.e.(x1  x2 )
*
df = min( n1  1, n2  1)
1   2
Parameter
Statistic
Standard Error
x1  x 2
pooled s.e.x1  x2   s p
where s p 
Confidence Interval

1 1

n1 n2
(n1  1)s12  (n 2  1) s 22
n1  n 2  2
Confidence Interval
x1  x2   t * pooled s.e.(x1  x2 )
df = n1  n2  2
161
Additional Notes
A place to … jot down questions you may have and ask during office hours, take a few
extra notes, write out an extra practice problem or summary completed in lecture, create
your own short summary about this chapter.
162
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